Subgroups — C₅₉×#33901 #28446010

Order 15104.

548 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3334	C₂	1	2	yes	maximal, complemented
2	C₅₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2232	C₂	1	4	yes	maximal, complemented
4	C₅₉×#2224	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2215	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₉×#366	C₂	1	2	yes	maximal, complemented
9	C₅₉×#547	—	2	2	no	—
10	C₂×D₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
12	C₅₉×#500	C₂²	1	2	yes	—
13	C₅₉×#480	—	2	4	no	—
14	C₅₉×#474	—	2	2	no	—
15	C₅₉×#466	C₂²	1	4	yes	—
16	C₄×D₈×C₅₉	C₂²	1	2	yes	—
17	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×#107×C₅₉	C₂²	1	4	yes	—
19	C₅₉×#366	C₂²	1	2	yes	—
20	C₅₉×#366	C₂²	1	4	yes	—
21	C₅₉×#366	C₂²	1	2	yes	—
22	C₅₉×#366	C₂²	1	1	yes	characteristic
23	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₅₉×#357	C₂²	1	2	yes	—
25	C₅₉×#357	C₂²	1	4	yes	—
26	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#41×C₅₉	—	4	2	no	—
29	C₂×D₈×C₅₉	C₂³	1	2	yes	—
30	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
31	C₄⋊Q₈×C₅₉	—	2	2	no	—
32	#123×C₅₉	—	4	4	no	—
33	C₂²⋊Q₈×C₅₉	—	4	2	no	—
34	#121×C₅₉	—	2	4	no	—
35	C₄×D₄×C₅₉	—	2	2	no	—
36	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
38	D₁₆×C₅₉	—	2	2	no	—
39	C₂×C₁₆×C₅₉	—	2	4	no	—
40	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
41	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
42	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
43	#107×C₅₉	—	2	4	no	—
44	#107×C₅₉	—	2	2	no	—
45	#107×C₅₉	C₂³	1	2	yes	—
46	#107×C₅₉	C₂³	1	4	yes	—
47	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
48	#102×C₅₉	—	2	2	no	—
49	#102×C₅₉	—	2	4	no	—
50	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
51	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	#41×C₅₉	—	8	8	no	—
53	C₂×Q₈×C₅₉	—	4	2	no	—
54	C₂×D₄×C₅₉	—	2	2	no	—
55	C₂×D₄×C₅₉	—	4	4	no	—
56	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	—	4	4	no	—
58	C₂²×C₄×C₅₉	—	4	2	no	—
59	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
60	D₈×C₅₉	—	2	2	no	—
61	D₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	—	2	2	no	—
63	C₂×C₈×C₅₉	—	2	2	no	—
64	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₅₉	—	2	2	no	—
68	C₄⋊C₄×C₅₉	—	2	4	no	—
69	C₄⋊C₄×C₅₉	—	4	4	no	—
70	C₄⋊C₄×C₅₉	—	4	2	no	—
71	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
72	C₂²⋊C₄×C₅₉	—	4	2	no	—
73	C₂²⋊C₄×C₅₉	—	4	4	no	—
74	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
75	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₅₉	—	2	2	no	—
77	C₁₆×C₅₉	—	2	1	no	—
78	C₁₆×C₅₉	#41	1	2	yes	complemented
79	C₂³×C₅₉	—	4	2	no	—
80	C₂³×C₅₉	C₂×D₈	1	2	yes	—
81	Q₈×C₅₉	—	8	4	no	—
82	D₄×C₅₉	—	4	2	no	—
83	D₄×C₅₉	—	4	2	no	—
84	D₄×C₅₉	—	8	8	no	—
85	D₄×C₅₉	D₁₆	1	4	yes	complemented
86	C₂×C₄×C₅₉	—	2	2	no	—
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	4	2	no	—
89	C₂×C₄×C₅₉	—	4	4	no	—
90	C₂×C₄×C₅₉	—	8	8	no	—
91	C₂×C₄×C₅₉	—	8	4	no	—
92	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₅₉	—	2	2	no	—
97	C₈×C₅₉	—	2	1	no	—
98	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
99	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
100	#33901	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₅₉	—	2	2	no	—
102	C₂²×C₅₉	—	2	2	no	—
103	C₂²×C₅₉	—	8	2	no	—
104	C₂²×C₅₉	—	8	2	no	—
105	C₂²×C₅₉	—	8	2	no	—
106	C₂²×C₅₉	C₂×D₁₆	1	4	yes	—
107	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
108	C₄×C₅₉	—	2	2	no	—
109	C₄×C₅₉	—	2	1	no	—
110	C₄×C₅₉	—	8	2	no	—
111	C₄×C₅₉	—	8	4	no	—
112	C₄×C₅₉	C₂×D₁₆	1	2	yes	—
113	C₄×C₅₉	#547	1	1	yes	characteristic
114	C₄×C₅₉	#578	1	1	yes	characteristic
115	#3334	C₂×C₅₉	1	2	yes	complemented
116	#2252	C₂×C₅₉	1	1	yes	characteristic, complemented
117	#2232	C₂×C₅₉	1	4	yes	complemented
118	#2224	C₂×C₅₉	1	2	yes	complemented
119	#2215	C₂×C₅₉	1	2	yes	complemented
120	C₄×D₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
121	D₄×C₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
122	C₂×#366	C₂×C₅₉	1	2	yes	complemented
123	C₂×C₅₉	—	2	4	no	—
124	C₂×C₅₉	—	16	2	no	—
125	C₂×C₅₉	#3334	1	1	yes	characteristic
126	C₂×C₅₉	C₂²×D₁₆	1	1	yes	characteristic
127	C₂×C₅₉	#3417	1	1	yes	characteristic
128	#547	—	2	2	no	—
129	C₂×D₁₆	C₂²×C₅₉	1	1	yes	characteristic
130	C₂²×C₁₆	C₂²×C₅₉	1	2	yes	—
131	#500	C₂²×C₅₉	1	2	yes	—
132	#480	—	2	4	no	—
133	#474	—	2	2	no	—
134	#466	C₂²×C₅₉	1	4	yes	—
135	C₄×D₈	C₂²×C₅₉	1	2	yes	—
136	C₈×D₄	C₂²×C₅₉	1	1	yes	characteristic
137	C₂×#107	C₂²×C₅₉	1	4	yes	—
138	#366	C₂²×C₅₉	1	1	yes	characteristic
139	#366	C₂²×C₅₉	1	2	yes	—
140	#366	C₂²×C₅₉	1	4	yes	—
141	#366	C₂²×C₅₉	1	2	yes	—
142	C₄⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
143	#357	C₂²×C₅₉	1	2	yes	—
144	#357	C₂²×C₅₉	1	4	yes	—
145	C₂²⋊C₁₆	C₂²×C₅₉	1	2	yes	—
146	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
147	C₅₉	#33901	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×#41	—	4	2	no	—
149	C₂×D₈	C₂³×C₅₉	1	2	yes	—
150	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
151	C₄⋊Q₈	—	2	2	no	—
152	#123	—	4	4	no	—
153	C₂²⋊Q₈	—	4	2	no	—
154	#121	—	2	4	no	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	2	4	no	—
158	D₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	4	no	—
160	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
162	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
163	#107	—	2	4	no	—
164	#107	—	2	2	no	—
165	#107	C₂³×C₅₉	1	2	yes	—
166	#107	C₂³×C₅₉	1	4	yes	—
167	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
168	#102	—	2	2	no	—
169	#102	—	2	4	no	—
170	C₂²⋊C₈	D₄×C₅₉	1	2	yes	—
171	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
172	#41	—	8	8	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	D₈×C₅₉	1	1	yes	characteristic
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
180	D₈	—	2	2	no	—
181	D₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
186	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	D₈×C₅₉	1	1	yes	characteristic
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	4	4	no	—
194	C₂²⋊C₄	D₈×C₅₉	1	2	yes	—
195	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#41×C₅₉	1	2	yes	complemented
199	C₂³	—	4	2	no	—
200	C₂³	C₂×D₈×C₅₉	1	2	yes	—
201	Q₈	—	8	4	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	8	8	no	—
205	D₄	D₁₆×C₅₉	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	8	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
213	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
214	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
219	C₈	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	8	2	no	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	2	no	—
225	C₂²	C₂×D₁₆×C₅₉	1	4	yes	—
226	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	8	2	no	—
230	C₄	—	8	4	no	—
231	C₄	C₂×D₁₆×C₅₉	1	2	yes	—
232	C₄	C₅₉×#547	1	1	yes	characteristic
233	C₄	C₅₉×#578	1	1	yes	characteristic
234	C₂	—	2	4	no	—
235	C₂	—	16	2	no	—
236	C₂	C₅₉×#3334	1	1	yes	characteristic
237	C₂	C₂²×D₁₆×C₅₉	1	1	yes	characteristic
238	C₂	C₅₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄×D₄×C₅₉ ◁ C₈×D₄×C₅₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅₉×#3334	C₂	semidirect
2. C₅₉×#2252	C₂	semidirect
3. C₅₉×#2232	C₂	semidirect
4. C₅₉×#2224	C₂	semidirect
5. C₅₉×#2215	C₂	semidirect
6. C₄×D₁₆×C₅₉	C₂	semidirect
7. D₄×C₁₆×C₅₉	C₂	semidirect
8. C₂×C₅₉×#366	C₂	semidirect
78. C₁₆×C₅₉	#41	semidirect
85. D₄×C₅₉	D₁₆	semidirect
100. #33901	C₅₉	direct (N × H)
115. #3334	C₂×C₅₉	semidirect
116. #2252	C₂×C₅₉	semidirect
117. #2232	C₂×C₅₉	semidirect
118. #2224	C₂×C₅₉	semidirect
119. #2215	C₂×C₅₉	semidirect
120. C₄×D₁₆	C₂×C₅₉	semidirect
121. D₄×C₁₆	C₂×C₅₉	semidirect
122. C₂×#366	C₂×C₅₉	semidirect
147. C₅₉	#33901	direct (N × H)
198. C₁₆	#41×C₅₉	semidirect
205. D₄	D₁₆×C₅₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₉×#3334	2
C₅₉×#2252	2
C₅₉×#2232	2
C₅₉×#2224	2
C₅₉×#2215	2
C₄×D₁₆×C₅₉	2
D₄×C₁₆×C₅₉	2
C₂×C₅₉×#366	2
#33901	59

Subgroups — C59×#33901 #28446010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33901 #28446010